#171962

order	512
nilpotency class	3
isoclinism	isoclinic class of #171725
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

(2,5)(3,7)(6,8)(9,10,14,17)(11,18,21,24)(12,16,13,22)(15,19,23,20)(30,37)(38,39);(9,11,14,21)(10,15,17,23)(12,20,13,19)(16,24,22,18)(25,26)(27,30)(28,35)(29,32)(31,36)(33,39)(34,37)(38,40);(1,2,4,6)(3,8,7,5)(25,27)(26,30)(28,33)(29,34)(31,38)(32,37)(35,39)(36,40);(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,18,23,24)(25,28,29,36)(26,31,32,35)(27,33,34,40)(30,38,37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 79158);